>>12505079The twenty-four pattern can be easily explained by the cancellation of the alternating sum binomial coefficients identity.
When you take the differences between each tier, you get a closed form formula which is a sum of the power four complex arithmetic progression, with alternating binomial coefficients.
Partition (into a term ) the root of each element of your arithmetic progression such that the smallest element is a complex number added to one, and the largest element is the same complex number added to six. Take the binomial expansion of each of the power four terms, distribute the alternating binomial coefficients, and regroup around the regular binomial coefficients.
The degree four complex number sum should cancel to zero because you can factor all of the non-alternating binomial coefficients out.
For the degree three complex number coefficient sum, you can factor out all of the non-alternating binomial coefficients out except for a degree one arithmetic progression. This progression is shifted such that you can’t cancel out with the alternating binomial coefficients. You can correct this by partitioning (into a term ) each element of the arithmetic progression into an integer plus one (or whatever the shift value is), then distributing over this partition. When you collect the terms for each partition, you will see that you can apply the alternating binomial sum to each, and cancel them both out to zero. The reason the shifting matters is because it shortens the sum by one element, and because it cancels out some of the denominators of the alternating binomial coefficients, producing a lower tier sum of alternating coefficients, but with a common factor (which can be distributed out).
This same procedure can be iteratively repeated on the other degree n complex number coefficient sums, but using different shift values on each iteration, but the cancellation identity stops working after a while.
